Algebra Examples

$\log (x + y) = \frac{1}{2} (\log (x) + \log (y)) + \log (2)$

Step 1

Simplify the right side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\log (x + y) = \frac{1}{2} \log (x) + \frac{1}{2} \log (y) + \log (2)$

Step 1.1.2

Combine $\frac{1}{2}$ and $\log (x)$ .

$\log (x + y) = \frac{\log (x)}{2} + \frac{1}{2} \log (y) + \log (2)$

Step 1.1.3

Combine $\frac{1}{2}$ and $\log (y)$ .

$\log (x + y) = \frac{\log (x)}{2} + \frac{\log (y)}{2} + \log (2)$

Step 2

Multiply each term in $\log (x + y) = \frac{\log (x)}{2} + \frac{\log (y)}{2} + \log (2)$ by $2$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\log (x + y) = \frac{\log (x)}{2} + \frac{\log (y)}{2} + \log (2)$ by $2$ .

$\log (x + y) \cdot 2 = \frac{\log (x)}{2} \cdot 2 + \frac{\log (y)}{2} \cdot 2 + \log (2) \cdot 2$

Step 2.2

Simplify the left side.

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Step 2.2.1

Move $2$ to the left of $\log (x + y)$ .

$2 \log (x + y) = \frac{\log (x)}{2} \cdot 2 + \frac{\log (y)}{2} \cdot 2 + \log (2) \cdot 2$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Cancel the common factor of $2$ .

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Step 2.3.1.1.1

Cancel the common factor.

$2 \log (x + y) = \frac{\log (x)}{2} \cdot 2 + \frac{\log (y)}{2} \cdot 2 + \log (2) \cdot 2$

Step 2.3.1.1.2

Rewrite the expression.

$2 \log (x + y) = \log (x) + \frac{\log (y)}{2} \cdot 2 + \log (2) \cdot 2$

Step 2.3.1.2

Cancel the common factor of $2$ .

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Step 2.3.1.2.1

Cancel the common factor.

$2 \log (x + y) = \log (x) + \frac{\log (y)}{2} \cdot 2 + \log (2) \cdot 2$

Step 2.3.1.2.2

Rewrite the expression.

$2 \log (x + y) = \log (x) + \log (y) + \log (2) \cdot 2$

Step 2.3.1.3

Move $2$ to the left of $\log (2)$ .

$2 \log (x + y) = \log (x) + \log (y) + 2 \log (2)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $2 \log (x + y)$ by moving $2$ inside the logarithm.

$\log ({(x + y)}^{2}) = \log (x) + \log (y) + 2 \log (2)$

Step 4

Simplify the right side.

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Step 4.1

Simplify $\log (x) + \log (y) + 2 \log (2)$ .

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Step 4.1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log ({(x + y)}^{2}) = \log (x y) + 2 \log (2)$

Step 4.1.2

Simplify each term.

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Step 4.1.2.1

Simplify $2 \log (2)$ by moving $2$ inside the logarithm.

$\log ({(x + y)}^{2}) = \log (x y) + \log (2^{2})$

Step 4.1.2.2

Raise $2$ to the power of $2$ .

$\log ({(x + y)}^{2}) = \log (x y) + \log (4)$

Step 4.1.3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log ({(x + y)}^{2}) = \log (x y \cdot 4)$

Step 4.1.4

Move $4$ to the left of $x y$ .

$\log ({(x + y)}^{2}) = \log (4 x y)$

Step 5

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

${(x + y)}^{2} = 4 x y$

Step 6

Solve for $x$ .

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Step 6.1

Move all terms containing $x$ to the left side of the equation.

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Step 6.1.1

Subtract $4 x y$ from both sides of the equation.

${(x + y)}^{2} - 4 x y = 0$

Step 6.1.2

Simplify each term.

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Step 6.1.2.1

Rewrite ${(x + y)}^{2}$ as $(x + y) (x + y)$ .

$(x + y) (x + y) - 4 x y = 0$

Step 6.1.2.2

Expand $(x + y) (x + y)$ using the FOIL Method.

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Step 6.1.2.2.1

Apply the distributive property.

$x (x + y) + y (x + y) - 4 x y = 0$

Step 6.1.2.2.2

Apply the distributive property.

$x \cdot x + x y + y (x + y) - 4 x y = 0$

Step 6.1.2.2.3

Apply the distributive property.

$x \cdot x + x y + y x + y \cdot y - 4 x y = 0$

Step 6.1.2.3

Simplify and combine like terms.

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Step 6.1.2.3.1

Simplify each term.

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Step 6.1.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x y + y x + y \cdot y - 4 x y = 0$

Step 6.1.2.3.1.2

Multiply $y$ by $y$ .

$x^{2} + x y + y x + y^{2} - 4 x y = 0$

Step 6.1.2.3.2

Add $x y$ and $y x$ .

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Step 6.1.2.3.2.1

Reorder $y$ and $x$ .

$x^{2} + x y + x y + y^{2} - 4 x y = 0$

Step 6.1.2.3.2.2

Add $x y$ and $x y$ .

$x^{2} + 2 x y + y^{2} - 4 x y = 0$

Step 6.1.3

Subtract $4 x y$ from $2 x y$ .

$x^{2} + y^{2} - 2 x y = 0$

Step 6.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.3

Substitute the values $a = 1$ , $b = - 2 y$ , and $c = y^{2}$ into the quadratic formula and solve for $x$ .

$\frac{2 y \pm \sqrt{{(- 2 y)}^{2} - 4 \cdot (1 \cdot y^{2})}}{2 \cdot 1}$

Step 6.4

Simplify.

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Step 6.4.1

Simplify the numerator.

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Step 6.4.1.1

Rewrite $4 \cdot 1 \cdot y^{2}$ as ${(2 \cdot 1 y)}^{2}$ .

$x = \frac{2 y \pm \sqrt{{(- 2 y)}^{2} - {(2 \cdot (1 y))}^{2}}}{2 \cdot 1}$

Step 6.4.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 2 y$ and $b = 2 \cdot 1 y$ .

$x = \frac{2 y \pm \sqrt{(- 2 y + 2 \cdot (1 y)) (- 2 y - (2 \cdot (1 y)))}}{2 \cdot 1}$

Step 6.4.1.3

Simplify.

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Step 6.4.1.3.1

Factor $2 y$ out of $- 2 y + 2 \cdot 1 y$ .

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Step 6.4.1.3.1.1

Factor $2 y$ out of $- 2 y$ .

$x = \frac{2 y \pm \sqrt{(2 y (- 1) + 2 \cdot (1 y)) (- 2 y - (2 \cdot (1 y)))}}{2 \cdot 1}$

Step 6.4.1.3.1.2

Factor $2 y$ out of $2 \cdot 1 y$ .

$x = \frac{2 y \pm \sqrt{(2 y (- 1) + 2 y (1)) (- 2 y - (2 \cdot (1 y)))}}{2 \cdot 1}$

Step 6.4.1.3.1.3

Factor $2 y$ out of $2 y (- 1) + 2 y (1)$ .

$x = \frac{2 y \pm \sqrt{2 y (- 1 + 1) (- 2 y - (2 \cdot (1 y)))}}{2 \cdot 1}$

Step 6.4.1.3.2

Add $- 1$ and $1$ .

$x = \frac{2 y \pm \sqrt{2 y \cdot (0 (- 2 y - (2 \cdot (1 y))))}}{2 \cdot 1}$

Step 6.4.1.3.3

Combine exponents.

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Step 6.4.1.3.3.1

Multiply $0$ by $2$ .

$x = \frac{2 y \pm \sqrt{0 y (- 2 y - (2 \cdot (1 y)))}}{2 \cdot 1}$

Step 6.4.1.3.3.2

Multiply $0$ by $y$ .

$x = \frac{2 y \pm \sqrt{0 (- 2 y - (2 \cdot (1 y)))}}{2 \cdot 1}$

Step 6.4.1.3.4

Factor $y$ out of $- 2 y - 1 \cdot 2 \cdot 1 y$ .

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Step 6.4.1.3.4.1

Factor $y$ out of $- 2 y$ .

$x = \frac{2 y \pm \sqrt{0 (y \cdot - 2 - 1 \cdot 2 \cdot (1 y))}}{2 \cdot 1}$

Step 6.4.1.3.4.2

Factor $y$ out of $- 1 \cdot 2 \cdot 1 y$ .

$x = \frac{2 y \pm \sqrt{0 (y \cdot - 2 + y (- 1 \cdot 2 \cdot 1))}}{2 \cdot 1}$

Step 6.4.1.3.4.3

Factor $y$ out of $y \cdot - 2 + y (- 1 \cdot 2 \cdot 1)$ .

$x = \frac{2 y \pm \sqrt{0 (y (- 2 - 1 \cdot 2 \cdot 1))}}{2 \cdot 1}$

Step 6.4.1.3.5

Multiply $- 1 \cdot 2 \cdot 1$ .

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Step 6.4.1.3.5.1

Multiply $- 1$ by $2$ .

$x = \frac{2 y \pm \sqrt{0 (y (- 2 - 2 \cdot 1))}}{2 \cdot 1}$

Step 6.4.1.3.5.2

Multiply $- 2$ by $1$ .

$x = \frac{2 y \pm \sqrt{0 (y (- 2 - 2))}}{2 \cdot 1}$

Step 6.4.1.3.6

Subtract $2$ from $- 2$ .

$x = \frac{2 y \pm \sqrt{0 y \cdot - 4}}{2 \cdot 1}$

Step 6.4.1.3.7

Combine exponents.

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Step 6.4.1.3.7.1

Multiply $0$ by $y$ .

$x = \frac{2 y \pm \sqrt{0 \cdot - 4}}{2 \cdot 1}$

Step 6.4.1.3.7.2

Multiply $0$ by $- 4$ .

$x = \frac{2 y \pm \sqrt{0}}{2 \cdot 1}$

Step 6.4.1.4

Rewrite $0$ as $0^{2}$ .

$x = \frac{2 y \pm \sqrt{0^{2}}}{2 \cdot 1}$

Step 6.4.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 y \pm 0}{2 \cdot 1}$

Step 6.4.1.6

$2 y$ plus or minus $0$ is $2 y$ .

$x = \frac{2 y}{2 \cdot 1}$

Step 6.4.2

Multiply $2$ by $1$ .

$x = \frac{2 y}{2}$

Step 6.4.3

Cancel the common factor of $2$ .

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Step 6.4.3.1

Cancel the common factor.

$x = \frac{2 y}{2}$

Step 6.4.3.2

Divide $y$ by $1$ .

$x = y$

Step 6.5

The final answer is the combination of both solutions.

$x = y$ Double roots